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The Paradoxical Success of Fuzzy Logic
 IEEE Expert
, 1993
"... Applications of fuzzy logic in heuristic control have been highly successful, but which aspects of fuzzy logic are essential to its practical usefulness? This paper shows that an apparently reasonable version of fuzzy logic collapses mathematically to twovalued logic. Moreover, there are few if any ..."
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Cited by 69 (1 self)
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Applications of fuzzy logic in heuristic control have been highly successful, but which aspects of fuzzy logic are essential to its practical usefulness? This paper shows that an apparently reasonable version of fuzzy logic collapses mathematically to twovalued logic. Moreover, there are few if any published reports of expert systems in realworld use that reason about uncertainty using fuzzy logic. It appears that the limitations of fuzzy logic have not been detrimental in control applications because current fuzzy controllers are far simpler than other knowledgebased systems. In the future, the technical limitations of fuzzy logic can be expected to become important in practice, and work on fuzzy controllers will also encounter several problems of scale already known for other knowledgebased systems. 1
Soft Computing: the Convergence of Emerging Reasoning Technologies
 Soft Computing
, 1997
"... The term Soft Computing (SC) represents the combination of emerging problemsolving technologies such as Fuzzy Logic (FL), Probabilistic Reasoning (PR), Neural Networks (NNs), and Genetic Algorithms (GAs). Each of these technologies provide us with complementary reasoning and searching methods to so ..."
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Cited by 51 (8 self)
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The term Soft Computing (SC) represents the combination of emerging problemsolving technologies such as Fuzzy Logic (FL), Probabilistic Reasoning (PR), Neural Networks (NNs), and Genetic Algorithms (GAs). Each of these technologies provide us with complementary reasoning and searching methods to solve complex, realworld problems. After a brief description of each of these technologies, we will analyze some of their most useful combinations, such as the use of FL to control GAs and NNs parameters; the application of GAs to evolve NNs (topologies or weights) or to tune FL controllers; and the implementation of FL controllers as NNs tuned by backpropagationtype algorithms.
Selforganized fuzzy system generation from training examples
 IEEE Trans. Fuzzy Syst
, 2000
"... Abstract—In the synthesis of a fuzzy system two steps are generally employed: the identification of a structure and the optimization of the parameters defining it. This paper presents a methodology to automatically perform these two steps in conjunction using a threephase approach to construct a fu ..."
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Cited by 23 (10 self)
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Abstract—In the synthesis of a fuzzy system two steps are generally employed: the identification of a structure and the optimization of the parameters defining it. This paper presents a methodology to automatically perform these two steps in conjunction using a threephase approach to construct a fuzzy system from numerical data. Phase 1 outlines the membership functions and system rules for a specific structure, starting from a very simple initial topology. Phase 2 decides a new and more suitable topology with the information received from the previous step; it determines for which variable the number of fuzzy sets used to discretize the domain must be increased and where these new fuzzy sets should be located. This, in turn, decides in a dynamic way in which part of the input space the number of fuzzy rules should be increased. Phase 3 selects from the different structures obtained to construct a fuzzy system the one providing the best compromise between the accuracy of the approximation and the complexity of the rule set. The accuracy and complexity of the fuzzy system derived by the proposed selforganized fuzzy rule generation procedure (SOFRG) are studied for the problem of function approximation. Simulation results are compared with other methodologies such as artificial neural networks, neurofuzzy systems, and genetic algorithms. Index Terms—Function approximation, fuzzy system design, generation of membership functions and rules. I.
Adapting the Gain of an FLC with Genetic Algorithms
 International Journal of Approximate Reasoning
, 1994
"... Fuzzy Logic Controllers are knowledgebased systems, incorporating human knowledge into their Knowledge Base through Fuzzy Rules and Fuzzy Membership Functions. The definition of these Fuzzy Rules and Fuzzy Membership Functions is generally affected by subjective decisions, having a great influence ..."
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Cited by 19 (2 self)
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Fuzzy Logic Controllers are knowledgebased systems, incorporating human knowledge into their Knowledge Base through Fuzzy Rules and Fuzzy Membership Functions. The definition of these Fuzzy Rules and Fuzzy Membership Functions is generally affected by subjective decisions, having a great influence over the performance of the Fuzzy Controller. In some cases, the Membership Functions are defined within a normalized interval, and the Knowledge Base includes a set of scaling functions to convert the input variables from its real value to a normalized one, and the normalized outputs to its real value. Different works have proposed the application of genetic strategies, with a learning purpose, to the Knowledge Base of Fuzzy Controllers. The learning is usually centered on the Membership Functions and/or the Rule Base, using fixed (predefined) scaling functions. In this paper, the evolution is applied to modify the gain of the controller (by modifying the scaling function of each input or o...
Stability of interpolative fuzzy KH controllers
, 2002
"... The classical approaches in fuzzy control (Zadeh and Mamdani) deal with dense rule bases. When this is not the case, i.e. in sparse rule bases, one has to choose another method. Fuzzy rule interpolation (proposed first by Koczy and Hirota [15]) offers a possibility to construct fuzzy controllers (KH ..."
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Cited by 13 (7 self)
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The classical approaches in fuzzy control (Zadeh and Mamdani) deal with dense rule bases. When this is not the case, i.e. in sparse rule bases, one has to choose another method. Fuzzy rule interpolation (proposed first by Koczy and Hirota [15]) offers a possibility to construct fuzzy controllers (KH controllers) under such conditions. The main result of this paper shows that the KH interpolation method is stable. It also contributes to the application oriented use of BalazsShepard interpolation operators investigated extensively by researchers in approximation theory. The numerical analysis aspect of the result contributes to the wellknown problem of finding a stable interpolation method in the following sense.
Investigation of Fuzzy Rule Interpolation Techniques and the Universal Approximation Property of Fuzzy Controllers
"... Fuzzy control is the most successfull application area of fuzzy theory. The advantage of fuzzy controllers against conventionel ones is that, they can by used for modelling systems with complicated (non linearizable) or unknown behaviour by means of linguistic variables anf fuzzy Ifthen rules. Late ..."
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Cited by 4 (1 self)
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Fuzzy control is the most successfull application area of fuzzy theory. The advantage of fuzzy controllers against conventionel ones is that, they can by used for modelling systems with complicated (non linearizable) or unknown behaviour by means of linguistic variables anf fuzzy Ifthen rules. Later on, the first approximating model can be tuned to obtain appropriate result. However, an essential problem of these algorithms is that their time complexity grows exponentially with the number of input variables. Fuzzy rule interpolation methods are one of the technique developed to reduce the complexity of fuzzy reasoning approaches. Purposes of this work are the following:  Modification of the first published KóczyHirota (KH) interpolation method to alleviate the socalled abnormal conclusion while maintaining its advantageous complexity behaviour.  Investigation of the mathematical stability of the KHmethod.  Examination of the universal approximation property of certain fuzzy controllers. A modification of the original KH approach was proposed, whose main idea is the following. The consequent fuzzy sets are transformed by a proper coordinate transformation to such a space where the convexity of these consequents excludes abnormality of the conclusion. After the conclusion is calculated in this space, the inverse of the aforementioned transformation is used to obtain the corresponding conclusion in the original output space. The proposed method is closed for convex and normal fuzzy sets (Theorem 2.1). The new interpolation method was compared with the KHapproach one in several aspects. It was investigated how the proposed method differs form linear between characteristic points, and finally a comparison among the main interpolation techniques is given with respect to the relation of the observation's and conclusion's fuzziness. It was proven that the inputoutput function of the KH interpolation converges uniformly to the arbitrary approximated continuous function if the measurement points are uniformly distributed on the domain. A generalization of this theorem is also given for a wider class of interpolatory operators. It was pointed out that the stability of the wellknown Shepardinterpolation (investigated extensively by approximation theorists) is can be derived from the one of the KH interpolation. The third main statement characterizes a set of certain type fuzzy controllers with bounded number of rules concerning the universal approximation property. As a generalization of Moser's result, it was shown that this property does not hold for the set of Tcontrollers (which includes Sugeno, TakagiSugeno, TakagiSugenoKang inference methods) if the number of rules is prerestricted, although that is a considerable practical limitation. It contradicts to those statements which state that fuzzy controllers are universal approximators, i.e., they lie dense in the space of continuous functions.
A New Scheme for an Automatic Generation of MultiVariable Fuzzy Systems
 Fuzzy Sets Syst
, 2001
"... We present a new novel method of automatically generating a multivariable fuzzy inference system from given sample sets. We first decompose the sample set, say , into a cluster of sample sets associated with the given input variables, then compute the associated fuzzy rules and membership functions ..."
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Cited by 1 (1 self)
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We present a new novel method of automatically generating a multivariable fuzzy inference system from given sample sets. We first decompose the sample set, say , into a cluster of sample sets associated with the given input variables, then compute the associated fuzzy rules and membership functions for each variable, independent of the other variables, by solving a single input multiple outputs fuzzy system extracted from the set cluster. The resulting decomposed fuzzy rules and membership functions for all the variables are integrated back into the fuzzy system appropriate for the original sample set . Taking an advantage of the independence of the input variables in computing the decomposed systems, we show that the computational complexity of the multivariable system can in principle be reduced to that of a single variable if we can use a parallel processing multiCPUs system. We have verified our claim using an eight variable nonlinear function. Keywords Fuzzy system, Membership ...
Investigation of a New AlphaCut Based Fuzzy Interpolation Method
"... The first published result in fuzzy rule interpolation was the ccut based fuzzy rule interpolation, termed as KIIinterpolation, originally devoted for complexity reduction. Some deficiencies of this method was presented later, such as abnormal conclusion for certain configuration of the invol ..."
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The first published result in fuzzy rule interpolation was the ccut based fuzzy rule interpolation, termed as KIIinterpolation, originally devoted for complexity reduction. Some deficiencies of this method was presented later, such as abnormal conclusion for certain configuration of the involved fuzzy sets. This inspired several authors to propose various conceptually different fuzzy interpolation approaches, however, none of those algorithms has such a low computational complexity than the KHmethod.